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Abstract We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach spaces. An application to the inverse scattering problem with diffuse waves is described.
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Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves
Abstract This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important partial differential equations (PDEs) in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under theLpnorms, we study these quantities under the SobolevHsnorm, which associates with a general class ofL2-based function spaces. TheHsnorm has a natural spectral bias based on its definition in the Fourier domain: the cases < 0 biases towards the lower frequencies, while the cases > 0 biases towards the higher frequencies. We compare the stability estimates using differentHsnorms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures. We also provide numerical inversion examples to demonstrate the differences in the inverse scattering radius of convergence under different metric spaces.
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- Award ID(s):
- 1913129
- PAR ID:
- 10405606
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 39
- Issue:
- 5
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- Article No. 054005
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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